ineAodus balamInternational Centre for Theoretical Physics

I - • - . , , • •• . | A l . ,

SMR 1673/34

AUTUMN COLLEGE ON PLASMA PHYSICS

5-30 September 2005

Quantum Mechanics and Nonadiabaticity inCharged Particle Dynamics

Ram K. Varma

P.R.L., Ahmedabad, India

Sirads Cosjiera I I. M0I4 Trieste. Italy - Tel. +39 010 1210 I I I: Fax +39 040 224 163 - sci_info@i«p.i!. www.inp.it

QUANTUM MECHANICS and NON-ADIABATICITY inCHARGED PARTICLE DYNAMICS

Ram K. VarmaPhysical Research Laboratory, Ahmedabad

India

ROSENBLUTH SYMPOSIUM

Abdus Salam International Centre for Theoretical PhysicsTrieste, Italy

September 19, 2005

Marshall RosenbluthAugust 1965

ADIABATICITY IN CLASSICAL MECHANICS

Adiabatic Invariance of action:

If Hamiltonian for a classical system ( one-dimensional, forsimplicity) ,depends on a small parameter, s ,slow time variationof A

H = H(p,q;A)

Then well known that the action, I = § p dq, an Invariant

d I / dt = 0, I = const.

Termed ' Adiabatic' because invariance follows only under the condition

8 = T d X/dt« 1, T = period of the motion

NONADIABATICITY: • Effect of the violation of the Adiabaticitycondition Td A/dt« 1/ e

CLASSICAL- CHARGED PARTICLE DYNAMICS

Gyro-action as an Adiabatic Invariant:

Ji — E ± / D , E ±= perpendicular energy,= eB /me, gyro-frequency

Under the'adiabaticity condition: hJt&Uk w»ud t81 = (1 / D2)(dD / d t ) « 1, for time variation, LP O-J ^ t fe x = (v± /12 B)(d B / dx) « 1, for spatial variation

Change in \i, if these conditions NOT fulfilled Non-adiabatic change

A (I ~ exp [-1/ s ]It is non-expandible in the small 'adiabaticity parameter'. Belongs to the

singular perturbation, and asymptotic phenomena ^Non-adiabatic effects— analogous to quantum effects!!! UK B w>

/ IThe adiabaticity parameter s has a role analogous to that of h in Q.M * PpHas quantum effect like the tunneling also goes as ~ exp [- |S| / h]

Non-adiabaticity to 'adiabatic motion' as quantum effects to classical dynamics

ADIABATIC INVARIANTS IN OLD QUANTUM THEORY

"Adiabatic Invariants" quantized in old quantum theory: M

n= action in units of h

Action j> p dq remains conserved as long as the particle stayed in a given orbit

Transition between levels • alters 'action' by a discrete ammout ^%^nh «(• Nonadibaticity-quantum mechanically

Adiabaticity condition, quantum mechanically:

The condition that the particle stays in the same orbit n against the perturbingpotential V is

t /(En- E n+1)[d V /dt] « 1

En, E n+1 energies of neighbouring levels; T, the period of the motion

This implies that the 'change of the perturbing potential in a period of the motionMuch less than the difference of energy between neighbouring levels

Charged Particle Dynamics in Quantum mechanics

Quantum mechanically charged particle in a magnetic field described bya wave function:

¥ N k (x, z) = H N(x) e»*

HN (x) = Harmonic oscillator wave function for the Landau level N *

e IKZ = plane w a ve along the magnetic field with k = mv /n

Eigenvalues: E N = (N+l/2) h £1, for the Landau levelsEz = (h k)2 / 2m, for the plane wave along z

r\ x 1 • 11 .1.1 X- *J *.*r- J Landau levels N

Quantum mechanically, the gyro-action identified as

Adiabatic Invariance of |i ^ • Adiabatic Invariance of N

Quantum mechanically, the adiabaticity condition' gives here[ x / ( E N - E N + 1 ) d V / d t « l

T = period of motion = 2 n /£l, for 4gyro-motion, E N= NhH, E N+1= (N+l)neighbouring levels, V= h £1, transition inducing potential in the present case

With this the above condition reduces to the classical case:( 2 n IQ2 )dQ / dt

Object and Focus of the Presentation

To show that:(i) Nonadiabatic effects in charged particle dynamics

are not just 'Quantum-like" but are indeed real quantum effects whichmanifest in Macroscopic dimensions in the correspondence limit.

(ii) These non-adiabatic effects are found to have unusual and novelmanifestations in the form of one-dimensional 'matter wave interference'effects, with matter wave length typically ~ 5 cm. This may sound quiteastonishing. But I shall show experimental evidence to that effect

(III) Even more astonishing, motivated by our theoretical prediction,we have detected the presence of a curl-free vector potential, inmacroscopic dimensions, in the spirit of the Aharonov-Bohm effect.

Finally (iv) nonadiabatic loss of charged particles from 'Adiabatic Traps'and multiplicity of 'residence times' observed 20 years back have beenIdentified as of 'quantum origin' in the correspondence limit

'Transition Amplitude' Wave Generated due to Scattering bya Center- a Simple Derivation

Charged Particle Moving in a uniform magnetic field:

Recall, the complete wave function of a charged particle in aUniform magnetic field, including the 'plane wave' along magnetic fieldgiven by

' z ) N( X ) l

Before scattering

z)

After scattering

Cik'z

S N' k'

N —-:y.»»>:>:y. Y. >:»>: ;-x-»>:>vxy. *»»:-x*:w

Landau levelsN

N+l+

- +

NONADIBATICITY IN CLASSICAL AND QUANTUM DESCRIPTIONS

Classically, nonadiabaticity at the end of a perturbation operation, is describedby one number A\i - AN h,

(i) AJI - exponentially small if magnetic field B an analytic function of time orspace. A\i ~ exp[-l/e]

(ii) A i, finite, if B or its derivatives undergoes a step function change.

Quantum mechanically, transitions across levels, whereby the (Landau)quantum number changes by AN, signifies nonadiabaticity.

Described by a set of " Transition Amplitudes":

9 (N\ N) = 9 (N, n), fromN—•N^N + n, n = l , 2 , 3 , ...

As a consequence of a perturbation, which may be(i) Magnetic field inhomogeneity or time dependence, which is NONADIBATIC(ii) Any external perturbation V(x) causing the transition N-*N'= N + n

Thenq> ( N \ N) = Jdx HN,(x) V(x) HN(x) = 9 (N, n), n=l, 2, 3...

These " Transition Amplitudes" specify QUANTUM NONADIBATICITY

Now the transition amplitudes <£(N, N ' , k) for 'transition' from

$(N, N', k ) =

—

=

Jdx

exp[exp[ -i

*N,k,(x, z) V(x)

(k'-k) z] Jdx H(k'-k) z] cp (N\

1 Nk VA> ^

fN<x) V(x)N)

H N( X )

Note that this expression represents a plane wave with the 'wave number'(k'-k) along the z-direction— along the magnetic field.

This is designated as the " Transition Amplitude" wave

To evaluate (k'-k) use the total energy conservation:

(hk')2/ 2m + N' hfl = (h k)2/2m + Nh Q

This yields: (k' - k) ^ (N- N') Q / (h k/m) = - (n D, I v)

(k'- k) « k assumed so that (k'+ k) ^ 2 k; h k/ m = v, velocity along z

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The "transition amplitude wave" then has the form : W ,/ -f-

, n) = e i ^n v ) z (p (N, n) ^ ^ ^/i ^

This represents a plane wave along z, the direction of the magnetic field, \ uT /with wave length

An = 2 7i v/ n Q, n, denotes the harmonic number Reca11 that n is the

Landau level intervalsubtending the

For typical laboratory conditions: transition.Energy E = 1 keV, Magnetic field B= 100 g % n ^ £

Aj a 5 cm !!!, - the fundamental *•-Lt w~ "-*-*•>

Such long wave length "Matter Waves" !!! ???Would they really manifest in Nature ???

Does "Transition Amplitude Wave" have a physically observableconsequence in the above form??

WE SHALL SEE

A2 2.5 cm —the first harmonic

11

A GENERAL FORMALISM for TRANSITION AMPLITUDE:Equation governing the evolution of 'transition amplitude'; the followinSchrodinger-like equation:* • - I r H ft e A

L n 5 t 2m

f4 7f ^ ^ = transition amplitude' across the Landau quantum number internal ,\* If A N= n, and /i= N h, the initial gyro-action of the quantum state N iP%p) . *" » /T / " ' For typical laboratory conditions (E ~ 1 keV, B ~ 100 g), N ~ 10 8 !r) j ^ l ^fr&TV

o.iQ* A z = z - c o m p o n e n t o f t h e c u r l - f r e e v e c t o r p o t e n t i a l A , w h i c h c a n i n d u c e t r a n s i t i o n l^^ U, 11 a c r o s s n = l , 2 , 3 . . . . , c/f / •

This equation offers the possibility of detecting the presence of ^_ \ J'A z, through one-dimensional interference effects in macroscopic dimensions Ji *' n

References with these developments: ^

1. R.K.Varma, Phys. Rev. E 64, 036608 (2001), Erratum E 65 019904(This reference contains the derivation of this equation from 'quantum mechanics) ' i«' f ij

2 . R.K.Varma, Phys. Reports 378, 301-434 (2003) ^(This reference reviews the work on nonadiabaticity, and the above development J w /including all the associated experimental work.) uj> A

fi- 42

Consider a schematic representation of an experiment as below:Transition amplitude wave generated at A

Source S T m^ ^veiling t 0 ^ e P*ate p i 17

Electrongun

E Transition amplitude wave generated atthe grid G and travelling to the plate P

A, Anode M a 8 n e t * c ^ e ^ * m e s

GridG* "" Plate P

Electrons propagate from the source S, the 'electron gun' travelling along themagnetic field B

(i) Primary de Broglie electron wave gets scattered near the anode Aby the E± in the gun-region.

Ej_ 'kicks' the electron up the higher Landau levels.Generates "Transition Amplitude Wave" originating at A, the anode

(ii) The unscattered primary wave amplitude reaching the grid G, is scatteredby the grid wires, generating another " Transition Amplitude Wave"originating at G

Important: Transition-amplitude-waves do not exist a priori, but are generatedat the scattering centers, e.g. the anode A, grid G, and plate surface P

13

"Transition-amplitude-wave" amplitude ip^ originating at the Anode Aand reaching the plate P:

t//AP = A exp[ i K (z p - z A)] , K= Q I v

"Transition-amplitude-wave" amplitude i//GP originating at the grid GAnd reaching the plate P:

i//GP = A' exp [ i K ( z p - z G) ] , K =

Total amplitude of the " Transition-amplitude-wave" at the plate P

= A exp [i K (z p - z A)] + A exp [i K ( z P - z G)]Assuming A - A'for simplicityIntensity at the plate:

I^PI2 = I^GP +^AP 12 = 2 |A|2 [ 1 + cos K LAG]

Interference maxima at:

Note that the sweep of K=£11 v, would cause theOscillation of plate currentAt the "frequency" LAG

j , 3 K=fi/vLAG = Anode grid distance (z A-z G)

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LAG=anode is the Anode-Grid distance. If grid G is close to the plateP, so that LGP « L AG , and L ^ = LAG

Then using K = DI v, this gives:

=l, 2, 3

As the condition for interference maxima involving " macroscopicMatter waves" arising from the transition amplitudes

This constitutes the PREDICTION of the above formalism on the existenceof one-dimensional 'macroscopic matter waves' with the wave length

X =2nv/Q, for the 'fundamental'X n = 2TT v / n Q, for the n* harmonic

Above condition implies that (L M/ X) = j , - an integral number of wave lengthsin the length L AP

Recall Xx = 5 cm, for E = 1 keV, B= 100 g

Experimental check ??

15

To check the above PREDICTIONS, the following experiment carried out:

Source S

Transition amplitude wave generated at Aand travelling to the plate P

Electrongun

hTraansition amplitude wa\je generated atthe grid G and travelling t<s the plate P

7777 ' A , Anode '""""* Magnetic fieldGrid

71

Plate P

The electrons from the source shot along magnetic field, and collected atthe plate P, as the CATHODE VOLATGE (that is electron energy)

IS SWEPT jr

What kind of plate current response one would expect ?? «hn V

Platecurrent

U;

Y

lc

)Y)

e-TthV.rY

Lth

Electron energy

tf. bJUvun •'Mr

716

Observed plate current as a function of the Cathode Voltage:

(a)

(b)

(c)

^M

r

Cathode Voitage{V)«oo tot

A f/"N. Qwi

VV-- -\/\A<y

A

PIM*

Si la

Fig. 10

Plate current in nano-amperes

(C)

All figures (a), (b), (c) Magnetic fieldB= 69 gauss. LAP= 51 cm

Grid-plate separation LGP:

(a) 2 cm, (b) 4 cm, (c) 6 cm

NOTICE, the 'totally unexpected'nature of the oscillatory plate currentresponse.

There are sharply defined maximaand minima.Note the progressive formation of"beat-like" structures from (a)-(c)

If they represent interference 'maximaand minima, they ought to fit the relationL2 J

Recall that the oscillation "frequency" wouldCorrespond to the distance L ^

17

TabOnergy peak positions

Table showing the fit for Fig (a)i} Quantum mimbec

corresponding to the cuwe of Fig. 10beam, velocity

I2345

2447

2m?173,3144?126.7

(an™1)

0,1975

021580.23560,25610.2?5<S.

faiB

•}

It!II121314

" idenfilmi, /.,=.-ambient mat

(on)

5tU

5 Li50J50J5OS

the plate-gun.;netjc field, O r

Peak No.

6?

9

IiW, dedu©2id from.

11.0-

96,785.676.669,0

he gjriBfiB

k=Q?2

(K2954-a^is03348

0.1534-0.172

the ^Lilianquency, and.

•

1516171.8\3

Stdisd. = 2Kb,

n the ©tectran

(era.)

5IJ" 5§,,7

5§J5#,f51J.

jietlc field 0 ~ ^9 g, LP = 51, and. tvera^ £ « ~;;cm

The value of L p = 50.8 cm deduced from observed data using the relationQ L p = 2 IT j v, matches closely with the value L p = 51 cm actually used in theexperiment signifies agreement with the theory

Furthermore, various interference peaks have been identified as j = 10, 11,...16, 17, 18, 19

This establishes the existence of Macroscopic matter wavesas also a manifestation of Quantum non-adiabaticity

18

1

A. Ito and Z. YoshidaPhys. Rev. E 63, 026502, (2001)

Experimental curve withB= 99.0 g, and L = 24 cm.

Notice the large amplitude of thedips inthe plate current profile- 60 %

Numerical simulation based on "resonanceProduction" of secondary electrons

Authors' quotes:

~ii! the simulation -Fig. 7!,with a reasonablesecondary electron yield, gives only small andnarrow peaks, while in the experiment moredrastic changes were observed.'"

In our experiment where the cathodevoltage is swept, the 'resonance

Production' mechanism would not bepossible.

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Existence of Matter Wave "Beats"— as a further confirmatory evidence

When experiment carried out with grid separated from the plate by a finitedistance, the cathode-voltage ( electron energy) sweep yields 6 beats' in theplate current:

Note the Matter wave (macroscopic)beats observed:

Grid-plate distance D = 10 cmAnode-plate distance L^ = 51 cm

Cathode Voltage (V)

Magnetic field B : (a) 69 g (b) 135 g

Electron energy sweep E: 5 eV—•500 eV implies sweep of k = QI(2 n v) = 0.042 B /V1/2' from:

(a) 1.3-* 1.3X 10-2forB=69g

(b) 3.3-* 1.3X 10-2forB=135g

R.K.Varma, A.M. Punithavelu, and S.B. Banerjee Phys. Rev. E 65, 026503 (2002)

20

Fig. 2

0.04 0.06 0.12 0.14

Figure (b) replotted as a function of[E(eV)] <1/2> ~ k = (2TT fl)[2E /m]

Sweep of k in the cosine term cos k L,would cause the variationof the plate current with a "frequency" L

Figure shows two frequencies:0.08 0.10

E(eV)"1/2

(i) A high frequency "carrier", and (ii) a low frequency "beat- envelops"

The two "frequencies" in the experiment which produce the "beats" are(a) Anode-plate distance L M (b) Anode-grid distance L AG

If observed beats are "wave-beats", then beat frequency must be L Ap- L AG = LGP

and "beat-maxima" ought to fit the relation: n = 2TT] v

B«i.t No.

1.2 •34

Si (cV>

55-083.3

141.7283.0

OJ82O0.6682.0,51030358

ir (cm "' 1 j5

432

LG p= D -. / ^ (cm)

6.16.05. S35.58

M a c e t i c field i1? ••-• 13.5 g, O ---•• ft cn:i and average I, ~-~ 5.9 a n .

21

Observation of the Curl-free Vector Potential in the Macro-domain:[Varma, Punithavelu, and Banerjee Phys. Lett. A 303,114 (2002)]

Insert a Rowland-ring (producing a curl-free vector potential)in the path of electrons travelling along an external magnetic field •.>

From the Schrodinger-like equationThe condition for interference maximaIn the presence of the vector potential:

Jdz [ m v + (e/c) A z] = 2TTJ \IA i= 1 9 3

Transition amplitude wave generated atA and G. The path difference is thus L AG

This leads to:m v L AG

Plate current in nA, and solenoid current in A (ampere)0,7 1.4 2.1 2

sin 60 = 2-rrj \i

<&= flux enclosedin the ring solenoid

Plots show the variationof plate current with thevariation of current inthe ring solenoid, forvarious electron energies

22

Two peculiarities of this effect vis-a-vis, the quantum Aharonov-Bohm effect

(i) It is found to exist in the macro-domain in the deci-meter rangeas against the 'micro-domain' of the Aharonov-Bohm effect,

(ii) It exists in one-dimension as against the minimum of two dimensionsrequired for the A-B effect.

Both these features may cause some consternation :

Because (i) will appear to be in apparent contradiction with the classicalLorentz equation, andbecause (ii) is against the well held belief that the A-B effect is of topologicalOrigin, and must occur in no less than two dimensions.

The resolution of (i) should come through the realization that the effect is, in fact,a quantum effect pushed into the macro-domain, the domain also of the 'transitionamplitude' in the correspondence limit.

The resolution of (ii) should come from the realization that the "transitionamplitudes" which play the seminal role in the phenomena are generated only atthe position of scattering, and path differences are reckoned from those points.

The fact that this effect has indeed been observed means in view of (i) and (ii)above that it is indeed an unusual effect— in fact, an unusual quantum effect.

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What is the significance of these observed effects ??

One can view them from at least two angles:

1. From the Nonadiabaticity angle, one can say that:These effects are a " novel manifestations of Quantum Nonadiabaticityin the macrodomain"

2. From the Quantum Mechanical angle :These are totally unexpected and hitherto unfamiliar revelations ofQuantum Dynamics manifesting in the macrodomain, and ought toentail a revision of our water-tight classification that particle relatedquantum effects belong only to the micro-domain.

Finally, these results bring to focus, the so-far unrecognized importanceof the Quantum mechanical object: the "transition amplitude" , which propagatesfrom the point of its generation at a scattering center, and does not exist, a priori.It is this which describes the exotic phenomena discussed above.

The dynamics described by the transition amplitude involved here has been termedas the " Macro-quantum Dynamics" and has been reviewed from both the aboveangles in

R. K. Varma Phys. Reports 378, 301- 434, (2003)

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! i f ? 5 ! • > • < > >

Marshall in 2002with Richard Morse

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MARSHALL ROSENBLUTH- A PERSONAL TRIBUTE

I would like to submit that the work that I am going to report has been guidedand inspired by the spirit of the iconic physicist, Marshall Rosenbluthin whose memory we are holding this symposium today.

It is not just plasma physics that I learnt from him as a student, but a wholemanner of thinking, choosing a problem, approaching a problem, and doingscience in general.

He was a charming person, with a sense of subtle and sharp humour, not tomention, the sharp scientific mind that he possessed. He thought through aproblem fast, keeping his coworkers always on their toes to work out a problembefore he would walk into their room with the solution. Those who worked withhim could not have remained untouched by the scientific fragrance that he exuded

He was an excellent teacher. The material that he gave us in the 1963 plasmaphysics course, I have preserved, for I still find some interesting things in it.

My bond with him was very personal, for I was one of his first students, and Iremember him with a great deal of affection, respect and fond memories of myassociation with him. In my tribute to him and to his memory I dedicate thiswork to him which I have carried out over the past three decades.

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